The proof is based on the abstract scheme developed by Sandier & Serfaty (see [1]}).
Equation (REF ) is the \(L^2\) -gradient flow of the Ginzburg-Landau energy \(F_\varepsilon \) .
More precisely, for any \(\varepsilon >0\)  and for any \(v\in H^1_{\operatorname{tan}}(M)\)  we let
\(\displaystyle E_\varepsilon (v):={\left\lbrace \begin{array}{ll}F_\varepsilon (v) -\pi n\vert \log \varepsilon \vert -n\gamma ,\,\,\,\,\,\,&\hbox{ if }D v\in L^2_{\operatorname{tan}}(M),\,\,\,\frac{1}{\varepsilon ^2}(1-\vert v\vert ^2)^2\in L^1(M)\\\\+\infty \,\,\,\,\,&\hbox{otherwise in }L^2_{\operatorname{tan}}(M).\end{array}\right.}\) 
Recall that \(E_\varepsilon \)  \(\Gamma \) -converges to \(W\)  and therefore, for any \(t\in [0,T)^*\) ,
\(\liminf _{\varepsilon \rightarrow 0}E_\varepsilon (u_\varepsilon (t)) \ge W(\mathbf {a}(t),\,\mathbf {d}(t),\,\xi (t)).\) 
We set \(X_\varepsilon := L^2_{\operatorname{tan}}(M)\) . We endow \(X_\varepsilon \)  with the norm \(\left\Vert  v \right\Vert _{X_\varepsilon }:=\frac{1}{\vert \log \varepsilon \vert ^{1/2}}\left\Vert  v \right\Vert _{L^2(M)}\) .
It is easy to check that the subdifferential of \(E_\varepsilon \)  (with respect to the scalar product in \(X_\varepsilon \) ) is singlevalued and is given by
\(\partial _{X_\varepsilon } E_\varepsilon (v) =\left| \log \varepsilon  \right|\left(-\Delta v + \frac{1}{\varepsilon ^2}(\vert v\vert _g^2 -1)v\right).\) 
Therefore the Ginzburg-Landau evolution (REF ) rewrites as the gradient flow
\({\left\lbrace \begin{array}{ll}\partial _t u_\varepsilon (t) + \partial _{X_\varepsilon }E_\varepsilon (u_\varepsilon ) = 0\\\\u_\varepsilon (0) = u_{\varepsilon }^0.\end{array}\right.}\) 
Equivalently, \(u_\varepsilon \)  is a curve of maximal slope for the energy \(E_\varepsilon \)  with respect to the slope \(\Vert  \partial _{X_\varepsilon }E_\varepsilon \Vert _{X_\varepsilon }\)  and thus it satisfies
\(\frac{1}{2}\int _{0}^t\left\Vert  \partial _t u_\varepsilon  \right\Vert _{X_\varepsilon }^2 \mathrm {d}s+ \frac{1}{2}\int _{0}^{t}\left\Vert  \partial _{X_\varepsilon }E_\varepsilon (u_\varepsilon (s)) \right\Vert ^2_{X_\varepsilon } \mathrm {d}s + E_\varepsilon (u_\varepsilon (t)) = E_\varepsilon (u_\varepsilon ^{0})\) 

for any \(t\in [0, \, T^*)\) .
Thanks to Proposition REF  (see in particular ())
we have
\(\liminf _{\varepsilon \rightarrow 0}\frac{1}{2}\int _{0}^t \left\Vert  \partial _t u_\varepsilon (s) \right\Vert ^2_{X_\varepsilon }\mathrm {d}s \ge \frac{\pi }{2} \int _{0}^t \left| \mathbf {a}^{\prime } \right|^2_{g}(s)\mathrm {d}s.\) 
On the other hand, Proposition REF  and Fatou's Lemma imply
\(\liminf _{\varepsilon \rightarrow 0}\frac{1}{2}\int _{0}^t \left\Vert  \partial _{E_\varepsilon } u_\varepsilon (s) \right\Vert _{X_\varepsilon }^2\mathrm {d}s \ge \frac{1}{2\pi }\int _{0}^t \left| \nabla _{\mathbf {a}}W(\mathbf {a}(s), \, \mathbf {d}, \, \xi (s)) \right|^2 \mathrm {d}s.\) 
Therefore, for any \(t\in [0,T^*)\)  we obtain
\( \begin{split}E_\varepsilon (u_\varepsilon ^0) -E_\varepsilon (u_\varepsilon (t)) &\ge \frac{\pi }{2} \int _{0}^t \left| \mathbf {a}^{\prime } \right|^2_{g}(s)\mathrm {d}s + \frac{1}{2\pi }\int _{0}^t \left| \nabla _{\mathbf {a}}W(\mathbf {a}(s), \, \mathbf {d}, \, \xi (s)) \right|^2 \mathrm {d}s +\mathrm {o}_{\varepsilon \rightarrow 0}(1) \\& \ge \int _{0}^t \sum _{j=1}^n \left(\frac{\mathrm {d}a_j}{\mathrm {d}t},\nabla _{a_j}W(\mathbf {a}, \, \mathbf {d}, \, \xi )\right)_g \mathrm {d}s+ \mathrm {o}_{\varepsilon \rightarrow 0}(1)\\& = W(\mathbf {a}(0),\,\mathbf {d},\,\xi ) - W(\mathbf {a}(t),\,\mathbf {d},\,\xi (t)) + \mathrm {o}_{\varepsilon \rightarrow 0}(1).\end{split}\) 
On the other hand, since the initial conditions are well prepared according to
(REF ), () and we have the \(\Gamma \) -convergence of \(E_\varepsilon \)  along \(u_\varepsilon (t)\)  (see (REF )),
we have
\(E_\varepsilon (u_\varepsilon ^0) -E_\varepsilon (u_\varepsilon (t)) \le W(\mathbf {a}(0),\,\mathbf {d},\,\xi ) - W(\mathbf {a}(t),\,\mathbf {d},\,\xi (t)) + \mathrm {o}_{\varepsilon \rightarrow 0}(1).\) 
Therefore, the inequalities in (REF ) become equalities.
In particular, we obtain
\(\frac{\pi }{2} \int _{0}^t \left| \mathbf {a}^{\prime } \right|^2_{g}(s)\mathrm {d}s + \frac{1}{2\pi }\int _{0}^t \left| \nabla _{\mathbf {a}}W(\mathbf {a}(s), \, \mathbf {d}, \, \xi (s)) \right|^2 \mathrm {d}s + W(\mathbf {a}(t),\,\mathbf {d},\,\xi (t)) = W(\mathbf {a}(0),\,\mathbf {d},\,\xi ),\) 
that is (REF ).
Moreover, we also obtain that
\(E_\varepsilon (u_\varepsilon (t)) = W(\mathbf {a}(t),\,\mathbf {d},\,\xi (t)) + \mathrm {o}_{\varepsilon \rightarrow 0}(1)\qquad \textrm {for any } t\in [0,T^*),\) 
namely that \(u_\varepsilon \)  remains well prepared along the evolution, or, in other words, that \(u_\varepsilon (t)\)  is a recovery sequence for the renormalized energy \(W\)  for any \(t\in (0,T^*)\) .
Finally, the fact that (REF ) is indeed equivalent to the differential formulation of the gradient flow (REF ) is standard. More precisely,
since the chain rule implies
\(\int _{0}^t \sum _{j=1}^n \left(\frac{\mathrm {d}a_j}{\mathrm {d}t},\nabla _{a_j}W(\mathbf {a}, \, \mathbf {d}, \, \xi )\right)_g\mathrm {d}s= W(\mathbf {a}(0),\,\mathbf {d},\,\xi ) - W(\mathbf {a}(t),\,\mathbf {d},\,\xi (t)),\) 

we obtain that for any \(t\in (0,T^*)\)  there holds
\(\frac{\pi }{2} \int _{0}^t \left| \mathbf {a}^{\prime } \right|^2_{g}(s)\mathrm {d}s + \frac{1}{2\pi }\int _{0}^t \left| \nabla _{\mathbf {a}}W(\mathbf {a}(s), \, \mathbf {d}, \, \xi (s)) \right|^2 \mathrm {d}s = \int _{0}^t \sum _{j=1}^n \left(\frac{\mathrm {d}a_j}{\mathrm {d}t},\nabla _{a_j}W(\mathbf {a}, \, \mathbf {d}, \, \xi )\right)_g\mathrm {d}s,\) 

which means
\(\int _{0}^t\left| \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbf {a}(s) +\frac{1}{\pi }\nabla _{a_j}W(\mathbf {a}(s), \, \mathbf {d}, \, \xi (s)) \right|_g^2 \mathrm {d}s= 0,\) 
and thus
\(\displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbf {a}(t) = -\frac{1}{\pi }\nabla _{a_j}W(\mathbf {a}(t), \, \mathbf {d}, \, \xi (t))\qquad \text{ for any }\quad t\in (0,T^*). \) 
